Finite Group DatabasebetaFinite Group Databasebeta| order | 22464 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | D13⋊C3×A4⋊S4 |
| Fitting, socle | C24×C13 ⋊ C2×C3×C3⋊D3 |
| derived | A42×C13 ⋊ C22×C3 |
| Sylow | C13 ⋊ C2×C3×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A42×C13 | C22×C3 |
| Fitting, socle | C24×C13 | C2×C3×C3⋊D3 |
| LCS | G ▷ A42×C13 | |
| UCS | C1 | |
1516 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1516 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,6,7,3,4,8)(5,12,11,9,10,13)(14,17)(15,16)(18,21,20);(1,2,5,4,8,11,10,12,13,7,6,9,3)(15,17)(20,21);(15,16,17)(19,20,21)112 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽13)×A4⋊S4 |
| maximal normal subgroup of | AGL(1,𝔽13)×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,6,7,3,4,8)(5,12,11,9,10,13)(14,17)(15,16)(18,21,20), (1,2,5,4,8,11,10,12,13,7,6,9,3)(15,17)(20,21), (15,16,17)(19,20,21));